If U ≠ ∅, which statement about A = Aᶜ is correct?
Answer and explanation
Correct answer: No such set A is possible.
A set and its complement are disjoint, so A ∩ Aᶜ = ∅. If A = Aᶜ, replacing Aᶜ by A would give A ∩ A = ∅, which means A = ∅. But then Aᶜ = U, and the equality A = Aᶜ would require ∅ = U. This contradicts U ≠ ∅. Equivalently, no nonempty universal set can be equal to its own complement. Therefore, option A is correct.
Frequently asked questions
What is the correct answer to this question?
No such set A is possible.
Why is this the correct answer?
A set and its complement are disjoint, so A ∩ Aᶜ = ∅. If A = Aᶜ, replacing Aᶜ by A would give A ∩ A = ∅, which means A = ∅. But then Aᶜ = U, and the equality A = Aᶜ would require ∅ = U. This contradicts U ≠ ∅. Equivalently, no nonempty universal set can be equal to its own complement. Therefore, option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
